Satellites KA- $12$ and SAL- $1$ have spotted a UFO. Scientists want to determine its distance from KA- $12$ so they can later determine its size. The distance between these satellites is $900 \text{ km}$. From KA- $12$ 's perspective, the angle between the UFO and SAL- $1$ is $60^\circ$. From SAL- $1$ 's perspective, the angle between the UFO and KA- $12$ is $75^\circ$. How far is the UFO from KA- $12$ ? Do not round during your calculations. Round your final answer to the nearest kilometer.
Solution: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $BC=d$. Because the interior angles of a triangle add to $180^\circ$, we know that $\angle B=45^\circ$. $A$ $B$ $C$ $75^\circ$ $60^\circ$ $45^\circ$ $900\text{ km}$ $d$ Since we are given one side length and all angle measures, we can use the law of sines. Using the law of sines $\begin{aligned} \dfrac{\sin(B)}{AC}&=\dfrac{\sin(A)}{BC}\\\\ \dfrac{\sin(45^\circ)}{900} &= \dfrac{\sin(75^\circ)}{d} \gray{\text{Substitute}} \\\\ d \cdot \sin(45^\circ) &= 900 \cdot \sin(75^\circ) \\\\ d &= \dfrac{900 \cdot \sin(75^\circ) }{\sin(45^\circ) } \\\\ d &\approx 1229 \,\text{km} \end{aligned}$ The answer The UFO is $1229 \,\text{km}$ from KA- $12$.